In a vehicle air conditioner for a so-called idle-stop vehicle, an evaporator having a cold storage function is often used for keeping cool feelings of a passenger at the time of temporary stop of the vehicle. Known as such evaporator having the cold storage function is one described in Patent Document 1, for example. The evaporator of Patent Document 1 is a type in which cold storage containers containing a cold storage material are disposed between multiple refrigerant tubes. Since the arrangement of the cold storage containers may cause retention and freezing of condensed water and deformation or damage of elements of the evaporator caused thereby, the cold storage container of the Patent Document 1 has multiple grooves or recess portions distributed with uniformity on its entire surface for promoting of flowing down of the condensed water due to gravity.